Combine and simplify the following radical expression:
$2 \sqrt{20}+8 \sqrt{45}-\sqrt{80}$
Answer (2)
The expression 2 20 + 8 45 − 80 simplifies to 24 5 after breaking down each radical. Each radical is simplified by identifying perfect squares, and like terms are combined afterwards. The final result is a single term that represents the original expression.
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Simplify each radical: 20 = 2 5 , 45 = 3 5 , 80 = 4 5 .
Substitute the simplified radicals into the expression: 2 ( 2 5 ) + 8 ( 3 5 ) − 4 5 .
Simplify the terms: 4 5 + 24 5 − 4 5 .
Combine like terms: ( 4 + 24 − 4 ) 5 = 24 5 .
The simplified expression is 24 5 .
Explanation
Understanding the Problem We are asked to combine and simplify the radical expression 2 20 + 8 45 − 80 . To do this, we need to simplify each radical term by factoring out perfect squares and then combine like terms.
Simplifying Radicals First, let's simplify each radical term individually:
20 = 4 ⋅ 5 = 4 ⋅ 5 = 2 5 45 = 9 ⋅ 5 = 9 ⋅ 5 = 3 5 80 = 16 ⋅ 5 = 16 ⋅ 5 = 4 5
Substituting Simplified Radicals Now, substitute the simplified radicals back into the original expression:
2 20 + 8 45 − 80 = 2 ( 2 5 ) + 8 ( 3 5 ) − 4 5
Simplifying Terms Next, simplify each term:
2 ( 2 5 ) = 4 5 8 ( 3 5 ) = 24 5 So the expression becomes:
4 5 + 24 5 − 4 5
Combining Like Terms Now, combine the like terms (terms with the same radical, 5 ):
( 4 + 24 − 4 ) 5 = 24 5
Final Answer Therefore, the simplified expression is 24 5 .
Examples
Radical expressions are useful in various fields, such as engineering and physics, for calculating lengths, areas, and volumes. For example, when calculating the diagonal of a square with side length s , the diagonal is s 2 . Simplifying radical expressions makes these calculations easier to work with in practical applications.
